McKean–Vlasov Process

In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself. The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. It is an example of propagation of chaos, in that it can be obtained as a limit of a mean-field system of interacting particles: as the number of particles tends to infinity, the interactions between any single particle and the rest of the pool will only depend on the particle itself.

Definition

Consider a measurable function $\sigma:\R^d \times \mathcal(\R^d)\to \mathcal_(\R)$ where $\mathcal(\R^d)$ is the space of probability distributions on $\R^d$ equipped with the Wasserstein metric $W_2$ and $\mathcal_(\R)$ is the space of square matrices of dimension $d$. Consider a measurable function $b:\R^d\times \mathcal(\R^d)\to \R^d$. Define $a(x,\mu) := \sigma(x,\mu)\sigma(x,\mu)^T$.

A stochastic process $(X_t)_$ is a McKean–Vlasov process if it solves the following system:

* $X_0$ has law $f_0$
* $dX_t = \sigma(X_t, \mu_t) dB_t + b(X_t, \mu_t) dt$

where $\mu_t = \mathcal(X_t)$ describes the law of $X$ and $B_t$ denotes a $d$-dimensional Wiener process. This process is non-linear, in the sense that the associated Fokker-Planck equation for $\mu_t$ is a non-linear partial differential equation.

Existence of a solution

The following Theorem can be found in.

Propagation of chaos

The McKean-Vlasov process is an example of propagation of chaos. What this means is that many McKean-Vlasov process can be obtained as the limit of discrete systems of stochastic differential equations $(X_t^i)_$.

Formally, define $(X^i)_$ to be the $d$-dimensional solutions to:

* $(X_0^i)_$ are i.i.d with law $f_0$
* $dX_t^i = \sigma(X_t^i, \mu_) dB_t^i + b(X_t^i, \mu_) dt$

where the $(B^i)_$ are i.i.d Brownian motion, and $\mu_$ is the empirical measure associated with $X_t$ defined by $\mu_ := \frac\sum\limits_ \delta_$ where $\delta$ is the Dirac measure.

Propagation of chaos is the property that, as the number of particles $N\to +\infty$, the interaction between any two particles vanishes, and the random empirical measure $\mu_$ is replaced by the deterministic distribution $\mu_t$.

Under some regularity conditions, the mean-field process just defined will converge to the corresponding McKean-Vlasov process.

Applications

* Mean-field theory
* Mean-field game theory
* Random matrices: including Dyson's model on eigenvalue dynamics for random symmetric matrices and the Wigner semicircle distribution

References

{{DEFAULTSORT:McKean-Vlasov process
Stochastic differential equations